Newform orbit 1216.2.n.b

• Label: 1216.2.n.b
• Level: $1216$
• Weight: $2$
• Character orbit: 1216.n
• Analytic conductor: $9.710$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1216.n (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.70980888579\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 304)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + ( - \zeta_{6} + 1) q^{3} + ( - 3 \zeta_{6} + 3) q^{5} + ( - 4 \zeta_{6} + 2) q^{7} + 2 \zeta_{6} q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 3 q^{5} + 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\)	\(191\)	\(705\)	\(837\)
\(\chi(n)\)	\(-1\)	\(\zeta_{6}\)	\(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

255.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

0.500000

−

0.866025i

0

1.50000

−

2.59808i

0

−

3.46410i

0

1.00000

+

1.73205i

0

639.1

0

0.500000

+

0.866025i

0

1.50000

+

2.59808i

0

3.46410i

0

1.00000

−

1.73205i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
76.f	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1216.2.n.b		2
4.b	odd	2	1		1216.2.n.a		2
8.b	even	2	1		304.2.n.a	✓	2
8.d	odd	2	1		304.2.n.b	yes	2
19.d	odd	6	1		1216.2.n.a		2
24.f	even	2	1		2736.2.bm.h		2
24.h	odd	2	1		2736.2.bm.g		2
76.f	even	6	1	inner	1216.2.n.b		2
152.l	odd	6	1		304.2.n.b	yes	2
152.o	even	6	1		304.2.n.a	✓	2
456.s	odd	6	1		2736.2.bm.g		2
456.v	even	6	1		2736.2.bm.h		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
304.2.n.a	✓	2	8.b	even	2	1
304.2.n.a	✓	2	152.o	even	6	1
304.2.n.b	yes	2	8.d	odd	2	1
304.2.n.b	yes	2	152.l	odd	6	1
1216.2.n.a		2	4.b	odd	2	1
1216.2.n.a		2	19.d	odd	6	1
1216.2.n.b		2	1.a	even	1	1	trivial
1216.2.n.b		2	76.f	even	6	1	inner
2736.2.bm.g		2	24.h	odd	2	1
2736.2.bm.g		2	456.s	odd	6	1
2736.2.bm.h		2	24.f	even	2	1
2736.2.bm.h		2	456.v	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1216, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - T + 1 \)
$5$	\( T^{2} - 3T + 9 \)
$7$	\( T^{2} + 12 \)
$11$	\( T^{2} + 12 \)